On transmission-type problems for the generalized Darcy-Forchheimer-Brinkman and stokes systems in complementary lipschitz domains in R3

The aim of our work is to give a well-posedness result for a boundary value problem of transmission-type for the nonlinear, generalized Darcy-Forchheimer-Brinkman and Stokes systems in complementary Lipschitz domains inR3. First, we introduce the Sobolev spaces in which we seek our solution, then we define the trace operators and conormal derivative operators that are involved in the boundary conditions of our treated problem. Next, we state a result that concerns the well-posedness of the transmission problem for the generalized Brinkman and Stokes system in complementary Lipschitz domains in R3. Afterwards, we state and prove an important lemma. Finally, we obtain our desired result by employing the well-posedness of the linearized version of our problem and Banach’s fixed point theorem.